The Etale Cohomology of Certain 'P'-Torsion Sheaves.

This dissertation generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincare characteristic of a constructible sheaf of F(,l)-modules on a smooth, proper curve over an algebraically closed field k of characteristic p > 0 as a sum of local terms and a global term, where l (NOT=) p. The primary focus is on removing the restriction on l. The previously known results were limited largely to calculations for p-torsion sheaves trivialized by p-extensions. We begin with a result similar to these, but using etale cohomology to give a unified proof for all primes l. In the remainder of this work, only p-torsion sheaves are considered. The main theoretical results occur in Chapter II. They are Theorem 2.1 and Corollary 2.7. The latter gives complete results on the Euler-Poincare characteristic of a constructible sheaf of F(,p)-modules on a complete, smooth curve over an algebraically closed field k of characteristic p > 0, when the generic stalk has rank p. Theorem 2.1 shows the existence of a certain short exact sequence of sheaves on X(,et), X a scheme of characteristic p, making possible the proof of 2.7 in the presence of Lemma 2.5. The sequence involves the tangent space at the identity of a finite, flat, height 1, commutative group scheme and the subsheaf fixed by the p-th power endomorphism; the latter turns out to be an etale group scheme. We conjecture the form of a general result for all constructible sheaves of F(,p)-modules over integral schemes in 2.4. Chapter III contains explicit computations of results from 2.7 for the case of sheaves over P('1). The topic of Chapter IV is the comparison of the p-ranks of abelian extensions of curves. It includes some examples of the calculation of the p-ranks of non-hyperelliptic curves.Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158840/1/8215009.pd